Theorem psubclssatN 40384

Description: A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
psubclssat.a	⊢ 𝐴 = (Atoms‘𝐾)
psubclssat.c	⊢ 𝐶 = (PSubCl‘𝐾)

Assertion

Ref	Expression
psubclssatN	⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ 𝐴)

Proof of Theorem psubclssatN

Step	Hyp	Ref	Expression
1		psubclssat.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
2		eqid 2737	. . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾)
3		psubclssat.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
4	1, 2, 3	psubcliN 40381	. 2 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → (𝑋 ⊆ 𝐴 ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑋)) = 𝑋))
5	4	simpld 494	1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  ‘cfv 6496  Atomscatm 39706  ⊥_𝑃cpolN 40345  PSubClcpscN 40377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5523  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-iota 6452  df-fun 6498  df-fv 6504  df-psubclN 40378

This theorem is referenced by:  pmapidclN  40385  psubclinN  40391  paddatclN  40392  pclfinclN  40393  poml6N  40398  osumcllem3N  40401  osumcllem9N  40407  osumcllem11N  40409  osumclN  40410